Quantum computing relies on the error-free operations of qubits.
Inevitably, qubits suffer from noise and need to be preserved using Quantum error correcting codes (QECC).
In the realm of quantum error correction, one of the elegant families of quantum codes is the Calderbank-Shor-Steane (CSS) codes named after their inventors [1]} whose structure motivates the construction of quantum codes using classical codes.
One of the important properties of CSS codes is that they consist of purely \(X\)  and \(Z\)  stabilizers [2]} which is particularly useful for the design of elegant decoders.
Some popular examples of CSS codes include the Shor's code, which encodes one logical qubit into nine physical qubits [3]}, and the Steane code, which has a higher coding rate and encodes one logical qubit into seven physical qubits [4]}. An interesting quantum code with an even better coding rate is the five-qubit code, i.e however non-CSS [5]}.
This non-CSS code with least number of physical qubits is capable of correcting any single-qubit error.
Motivation for the construction of non-CSS codes stems from Measurement-based quantum computing (MBQC) proposed by Raussendorf et al. [6]}. This scheme is shown to be viable for universal quantum computing.
It makes use of a resource state called cluster state consisting of entangled qubits, initially prepared in the \({+}\)  state followed by single qubit measurement.
The model of computation is therefore also called cluster state computation.
The success of this approach is based on the creative utilisation of qubit measurements. A 2D grid cluster state is shown in Fig. REF . This cluster state can be parameterized by mathematical structure given by \(G=(V,E)\) , where \(G\)  denotes the graph state associated with the cluster, \(V\)  indicates the set of vertices and \(E\)  represents the set of edges.
If one of the qubits in this cluster state is used as a message qubit then the resultant quantum state becomes a quantum error correcting code with one logical qubit.
